8,183 research outputs found
Fundamental Properties of Sum-Rank Metric Codes
This paper investigates the theory of sum-rank metric codes for which the
individual matrix blocks may have different sizes. Various bounds on the
cardinality of a code are derived, along with their asymptotic extensions. The
duality theory of sum-rank metric codes is also explored, showing that MSRD
codes (the sum-rank analogue of MDS codes) dualize to MSRD codes only if all
matrix blocks have the same number of columns. In the latter case, duality
considerations lead to an upper bound on the number of blocks for MSRD codes.
The paper also contains various constructions of sum-rank metric codes for
variable block sizes, illustrating the possible behaviours of these objects
with respect to bounds, existence, and duality properties
Generalized weights: an anticode approach
In this paper we study generalized weights as an algebraic invariant of a
code. We first describe anticodes in the Hamming and in the rank metric,
proving in particular that optimal anticodes in the rank metric coincide with
Frobenius-closed spaces. Then we characterize both generalized Hamming and rank
weights of a code in terms of the intersection of the code with optimal
anticodes in the respective metrics. Inspired by this description, we propose a
new algebraic invariant, which we call "Delsarte generalized weights", for
Delsarte rank-metric codes based on optimal anticodes of matrices. We show that
our invariant refines the generalized rank weights for Gabidulin codes proposed
by Kurihara, Matsumoto and Uyematsu, and establish a series of properties of
Delsarte generalized weights. In particular, we characterize Delsarte optimal
codes and anticodes in terms of their generalized weights. We also present a
duality theory for the new algebraic invariant, proving that the Delsarte
generalized weights of a code completely determine the Delsarte generalized
weights of the dual code. Our results extend the theory of generalized rank
weights for Gabidulin codes. Finally, we prove the analogue for Gabidulin codes
of a theorem of Wei, proving that their generalized rank weights characterize
the worst-case security drops of a Gabidulin rank-metric code
A Polymatroid Approach to Generalized Weights of Rank Metric Codes
We consider the notion of a -polymatroid, due to Shiromoto, and the
more general notion of -demi-polymatroid, and show how generalized
weights can be defined for them. Further, we establish a duality for these
weights analogous to Wei duality for generalized Hamming weights of linear
codes. The corresponding results of Ravagnani for Delsarte rank metric codes,
and Martinez-Penas and Matsumoto for relative generalized rank weights are
derived as a consequence.Comment: 22 pages; with minor revisions in the previous versio
On the similarities between generalized rank and Hamming weights and their applications to network coding
Rank weights and generalized rank weights have been proven to characterize
error and erasure correction, and information leakage in linear network coding,
in the same way as Hamming weights and generalized Hamming weights describe
classical error and erasure correction, and information leakage in wire-tap
channels of type II and code-based secret sharing. Although many similarities
between both cases have been established and proven in the literature, many
other known results in the Hamming case, such as bounds or characterizations of
weight-preserving maps, have not been translated to the rank case yet, or in
some cases have been proven after developing a different machinery. The aim of
this paper is to further relate both weights and generalized weights, show that
the results and proofs in both cases are usually essentially the same, and see
the significance of these similarities in network coding. Some of the new
results in the rank case also have new consequences in the Hamming case
Partitions of Matrix Spaces With an Application to -Rook Polynomials
We study the row-space partition and the pivot partition on the matrix space
. We show that both these partitions are reflexive
and that the row-space partition is self-dual. Moreover, using various
combinatorial methods, we explicitly compute the Krawtchouk coefficients
associated with these partitions. This establishes MacWilliams-type identities
for the row-space and pivot enumerators of linear rank-metric codes. We then
generalize the Singleton-like bound for rank-metric codes, and introduce two
new concepts of code extremality. Both of them generalize the notion of MRD
codes and are preserved by trace-duality. Moreover, codes that are extremal
according to either notion satisfy strong rigidity properties analogous to
those of MRD codes. As an application of our results to combinatorics, we give
closed formulas for the -rook polynomials associated with Ferrers diagram
boards. Moreover, we exploit connections between matrices over finite fields
and rook placements to prove that the number of matrices of rank over
supported on a Ferrers diagram is a polynomial in , whose
degree is strictly increasing in . Finally, we investigate the natural
analogues of the MacWilliams Extension Theorem for the rank, the row-space, and
the pivot partitions
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